Abstract
The complex Toda chain (CTC) is known to describe the N-soliton train interactions in adiabatic approximation for several nonlinear evolution equations: the nonlinear Schrödinger equation (NLS), the modified NLS, higher NLS. We briefly review and extend the results in order to treat several types of perturbations of the above-mentioned equations. In particular we consider analytically the effects of quadratic and periodic external potentials as perturbations. We also briefly analyze the Hamiltonian properties of the CTC.
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References
V. I. Karpman, V. V. Solov’ev. Physica D 3D, 487–502, (1981).
V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev. Phys. Rev. Lett. 77, 3943–3946 (1996).
V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov. Phys. Rev. E 55, No 5, 6039–6060 (1997).
J. M. Arnold. JOSA A 15, 1450–1458 (1998); Phys. Rev. E 60, 979–986 (1999).
V. S. Gerdjikov, E. G. Evstatiev, D. J. Kaup, G. L. Diankov, I. M. Uzunov. Phys. Lett. A 241, 323–328 (1998).
Y. Kodama, A. Hasegawa. IEEE J. of Quant. Electron. QE-23, 510–524 (1987).
V. S. Gerdjikov, M. I. Ivanov, P. P. Kulish. Theor. Math. Phys. 44, No.3, 342–357, (1980), (In Russian). V. S. Gerdjikov, M. I. Ivanov. Bulgarian J. Phys. 10, No.1, 13–26; No.2, 130–143, (1983), (In Russian).
V. S. Shchesnovich and E. V. Doktorov, Physica D 129, 115 (1999); E. V. Doktorov and V. S. Shchesnovich, J. Math. Phys. 36, 7009 (1995).
V. S. Gerdjikov, E. V. Doktorov, J. Yang. Phys. Rev. E 64, 056617 (2001).
V. S. Gerdjikov, I. M. Uzunov. Physica D 152–153, 355–362 (2001).
E. G. Evstatiev. N-soliton interactions of higher NLS equations. Diploma thesis, Sofia University (1995) (unpublished).
V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. I. Pitaevskii. Theory of solitons: the inverse scattering method (Plenum, N.Y.: Consultants Bureau, 1984).
L. A. Takhtadjan and L. D. Faddeev, Hamiltonian Approach to Soliton Theory. (Springer Verlag, Berlin, 1986).
S. V. Manakov. JETPh 67, 543 (1974); H. Flaschka. Phys. Rev. B9, n. 4, 1924–1925, (1974)
V. S. Gerdjikov. Complex Toda Chain — an Integrable Universal Model for Adiabatic N-soliton Interactions! In: Proc. of the workshop “Nonlinear Physics: Theory and Experiment.II”, Gallipoli, June–July 2002; Eds.: M. Ablowitz, M. Boiti, F. Pempinelli, B. Prinari. p. 64–70 (2003).
J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz Phys. Rev. E 63, 036612 (2001); Phys. Rev. Lett. 86, 1402–1405 (2001); J. C. Bronski, L. D. Carr, R. Carretero-Gonzales, B. Deconinck, J. N. Kutz, K. Promislow. Phys. Rev. E 64, 056615 (2001); F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, M. Salerno. Phys. Rev. A 64, 043606 (2001).
J. Moser, In Dynamical Systems, Theory and Applications. Lecture Notes in Physics, v. 38, Springer Verlag, (1975), p. 467. V. S. Gerdjikov, E. G. Evstatiev, R. I. Ivanov. The complex Toda chains and the simple Lie algebras — solutions and large time asymptotics J. Phys. A: Math & Gen. 31, 8221–8232 (1998).
V. S. Gerdjikov, A. Kyuldjiev, G. Marmo, G. Vilasi, Eur. Phys. J. B29, 177–181 (2002).
E. Doktorov. Private communication; see also E. V. Doktorov, N. P. Matsuka, V. M. Rothos. nlin.SI/0311010 (2003).
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Gerdjikov, V. (2004). On Modeling Adiabatic N-Soliton Interactions. In: Abdullaev, F.K., Konotop, V.V. (eds) Nonlinear Waves: Classical and Quantum Aspects. NATO Science Series II: Mathematics, Physics and Chemistry, vol 153. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2190-9_2
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DOI: https://doi.org/10.1007/1-4020-2190-9_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-2188-6
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